Hölder Continuity of Harmonic Quasiconformal Mappings

Abstract

We prove that for harmonic quasiconformal mappings $\alpha$-Hölder continuity on the boundary implies $\alpha$-Hölder continuity of the map itself. Our result holds for the class of uniformly perfect bounded domains, in fact we can allow that a portion of the boundary is thin in the sense of capacity. The problem for general bounded domains remains open.

This is an updated version of the same paper - added two references and a sentence explaining relevance of these two papers. The last version: corrected typos, restricted to dimension at least 3